L(s) = 1 | + (0.866 + 0.5i)2-s + (0.5 − 0.866i)3-s + (−0.500 − 0.866i)4-s − 2i·5-s + (0.866 − 0.499i)6-s + (−3.46 + 2i)7-s − 3i·8-s + (−0.499 − 0.866i)9-s + (1 − 1.73i)10-s + (−3.46 − 2i)11-s − 12-s − 3.99·14-s + (−1.73 − i)15-s + (0.500 − 0.866i)16-s + (1 + 1.73i)17-s − 0.999i·18-s + ⋯ |
L(s) = 1 | + (0.612 + 0.353i)2-s + (0.288 − 0.499i)3-s + (−0.250 − 0.433i)4-s − 0.894i·5-s + (0.353 − 0.204i)6-s + (−1.30 + 0.755i)7-s − 1.06i·8-s + (−0.166 − 0.288i)9-s + (0.316 − 0.547i)10-s + (−1.04 − 0.603i)11-s − 0.288·12-s − 1.06·14-s + (−0.447 − 0.258i)15-s + (0.125 − 0.216i)16-s + (0.242 + 0.420i)17-s − 0.235i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.543 + 0.839i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.543 + 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.605648 - 1.11443i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.605648 - 1.11443i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (-0.866 - 0.5i)T + (1 + 1.73i)T^{2} \) |
| 5 | \( 1 + 2iT - 5T^{2} \) |
| 7 | \( 1 + (3.46 - 2i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (3.46 + 2i)T + (5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-1 - 1.73i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-5 + 8.66i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 4iT - 31T^{2} \) |
| 37 | \( 1 + (-1.73 - i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-5.19 - 3i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (6 + 10.3i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + (-10.3 + 6i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1 - 1.73i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (6.92 + 4i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 2iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 4iT - 83T^{2} \) |
| 89 | \( 1 + (-1.73 - i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (8.66 - 5i)T + (48.5 - 84.0i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.38516686379216812754785710299, −9.620408432861726135263780124585, −8.812968544246956760293060853275, −7.967242877284151867198205796654, −6.60092293847460045573578207566, −5.89703178190644632425152677084, −5.15793714502192657356880132517, −3.84388913374310136613066332041, −2.57776087182548201477794171579, −0.58427601069559211659278518920,
2.77682253026464955026954085100, 3.20042096471468075968969185059, 4.30324775280008398674875661608, 5.32759007511666517854498648748, 6.76475613307462198498448976832, 7.45832661107519511001384848081, 8.589947577738212527438042482678, 9.708459303304471877678323976374, 10.39387809811603231696194845844, 11.03703227081171774483274474939